Calculus of Real and Complex Variables

4.2.3 A Physical Application, Work

One of the most important applications of line integrals is to the concept of work. I will specialize the
discussion to the case of a smooth curve. However, the general case is obtained by writing dγ in place of
γ′

(t)

dt. It is being assumed that the parametrizations considered are one to one on the interior of the
parameter domain.

You have a force field field F which acts on an object as it moves along the curve C, in the direction
determined by the given orientation of the curve which comes here from the given parametrization. From
beginning physics or calculus, you defined work as the force in a given direction times the distance
the object moves in that direction. Work is really a measure of the extent to which a force
contributes to the motion but here the object moves over a curve so the direction of motion
is constantly changing. In order to define what is meant by the work, consider the following
picture.

PICT

In this picture, the work done by a constant force F on an object which moves from the point
γ

(t)

to the point γ

(t+ h)

along the straight line shown would equal F ⋅

(γ(t+ h)− γ (t))

.
It is reasonable to assume this would be a good approximation to the work done in moving
along the curve joining γ

(t)

and γ

(t+ h)

provided h is small enough. Also, provided h is
small,

′
γ (t+ h)− γ(t) ≈ γ (t)h

where the wriggly equal sign indicates the two quantities are close. In the notation of Leibniz, one writes dt
for h and

′
dW = F(γ (t))⋅γ (t)dt

or in other words,

dW
---= F (γ(t))⋅γ′(t).
dt

Defining the total work done by the force at t = 0, corresponding to the first endpoint of the curve, to equal
zero, the work would satisfy the following initial value problem.

dW
-dt-= F(γ (t))⋅γ′(t),W (a) = 0.

This motivates the following definition of work.

Definition 4.2.10Let F

(γ)

be given above. Then thework done by this force field on an objectmoving over the curve C in the direction determined by the specified orientation is definedas

∫ b ∫
F (γ(t)) ⋅γ′(t)dt = F ⋅dγ
a C

where the functionγis one of the allowed parameterizations of C in the given orientation of C. In otherwords, there is an interval

[a,b]

and as t goes from a to b,γ

(t)

moves from one endpoint to theother.

When you have a curve and γ :

[a,b]

→ C and η :

[c,d]

→ C are two parametrizations such that γ−1∘η
is increasing, the two parametrizations give the same value for the work. This follows from
Theorem 4.1.3. You consider γ and γ∘

( −1 )
γ ∘ η

in that theorem. This says essentially that two
different parametrizations give the same value for the work if they go over the curve in the same
direction.

Example 4.2.11Say F ≡

(x,x + y,z − x)

. Find the work done on an object which moves overthe curve parametrized byγ